L(s) = 1 | + (−0.456 − 2.18i)5-s + 4.37i·7-s + 5.58i·11-s + 4.37·13-s − 5.58i·17-s + 4i·19-s + (−4.58 + 1.99i)25-s + 2.55i·29-s − 5.29·31-s + (9.58 − 1.99i)35-s + 2.55·37-s − 6·41-s − 11.1·43-s − 6.92i·47-s − 12.1·49-s + ⋯ |
L(s) = 1 | + (−0.204 − 0.978i)5-s + 1.65i·7-s + 1.68i·11-s + 1.21·13-s − 1.35i·17-s + 0.917i·19-s + (−0.916 + 0.399i)25-s + 0.473i·29-s − 0.950·31-s + (1.61 − 0.338i)35-s + 0.419·37-s − 0.937·41-s − 1.70·43-s − 1.01i·47-s − 1.73·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.450 - 0.892i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.450 - 0.892i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.208125035\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.208125035\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (0.456 + 2.18i)T \) |
good | 7 | \( 1 - 4.37iT - 7T^{2} \) |
| 11 | \( 1 - 5.58iT - 11T^{2} \) |
| 13 | \( 1 - 4.37T + 13T^{2} \) |
| 17 | \( 1 + 5.58iT - 17T^{2} \) |
| 19 | \( 1 - 4iT - 19T^{2} \) |
| 23 | \( 1 - 23T^{2} \) |
| 29 | \( 1 - 2.55iT - 29T^{2} \) |
| 31 | \( 1 + 5.29T + 31T^{2} \) |
| 37 | \( 1 - 2.55T + 37T^{2} \) |
| 41 | \( 1 + 6T + 41T^{2} \) |
| 43 | \( 1 + 11.1T + 43T^{2} \) |
| 47 | \( 1 + 6.92iT - 47T^{2} \) |
| 53 | \( 1 - 7.84T + 53T^{2} \) |
| 59 | \( 1 - 1.58iT - 59T^{2} \) |
| 61 | \( 1 - 10.5iT - 61T^{2} \) |
| 67 | \( 1 + 3.16T + 67T^{2} \) |
| 71 | \( 1 + 6.92T + 71T^{2} \) |
| 73 | \( 1 - 12iT - 73T^{2} \) |
| 79 | \( 1 - 5.29T + 79T^{2} \) |
| 83 | \( 1 + 7.16T + 83T^{2} \) |
| 89 | \( 1 - 2T + 89T^{2} \) |
| 97 | \( 1 + 11.1iT - 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.809440647373878952732269347199, −8.582614045534340920467802072966, −7.57181344505891577531751809814, −6.78030268166629494201826630878, −5.69003564846155454192107131455, −5.25946279492311483578990941845, −4.45935265908087174559311304201, −3.44067254180529420341981599715, −2.24039057789340731232075460272, −1.47042404211100835829295646285,
0.38849009286953441849852485164, 1.57039487008651901158980881101, 3.18218742723311018042917160468, 3.59667259589782566899619244285, 4.30568823364646889380327891392, 5.66231107722288531092105331992, 6.42986270797432740678268438837, 6.85151347333452014125193864695, 7.906898750382001970749752784786, 8.275115429768367435206475589821